Comparison between the zeroth-order Randić index and the sum-connectivity index

The zeroth-order Randić index and the sum-connectivity index are very popular topological indices in mathematical chemistry. These two indices are based on vertex degrees of graphs and attracted a lot of attention in recent years. Recently Li and Li (2015) studied these two indices for trees of order n. In this paper we obtain a relation between the zeroth-order Randić index and the sum-connectivity index for graphs. From this we infer an upper bound for the sum-connectivity index of graphs. Moreover, we prove that the zeroth-order Randić index is greater than the sum-connectivity index for trees. Finally, we show that R_{2, α}(G) is greater or equal R_{1, α}(G) (α ≥ 1) for any graph and characterize the extremal graphs.